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Extra info for An Introduction to Linear Analysis (Addison-Wesley Series in Mathematics)

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We can order the columns by the numerical values of the r-tuples thought of as binary representations of integers. Any other ordering would give an equivalent code. 1. Clearly each Ham(r,2) is a single-error-correcting code because the vectors of weight 1 have distinct syndromes, the columns of its parity check matrix. The general Hamming code can be easily shown to be a binary [2r — 1,2r — 1 — r, 3] code. There is an entirely analogous definition of a general Hamming code over GF(q); here we take one of the q — 1 nonzero multiples of a nonzero vector for a column of a parity check matrix of Ham(r, q).

For any a, by the division algorithm, we can write fix) = ix - a)bix) + rix) where the degree of rix) is less than 1. Hence rix) is a constant that must be 0 since ria) = 0. ■ Corollary. A polynomial over afieldF of degree n has at most n distinct roots. Just as we can consider integers modulo an integer, we can consider polynomials modulo some polynomial fix). As usual, our polynomials are all over the same field. We say that aix) is congruent to bix) modulo fix), denoted by aix) Ξ bix) (modulo fix)), iff there is a cix) with aix) = cix)fix) + bix).

1]. Theorem 17. If d is even, A(n - 1, d - 1) = Λ(η, d). Proof. Suppose that we have a set S of M vectors of length n - 1 whose minimum distance is d - 1. If two vectors are distance d - 1 apart, then one must have odd weight and the other even weight, since d - 1 is odd. Add an overall parity check to the vectors in 5. Then two vectors that were distance d - 1 apart will be at distance d from each other. Hence we have a set of M vectors of length n whose minimum distance is rf, showing that A(n - 1, d\)<>A{n,d\ Suppose now that we have a set of M vectors of length n and minimum distance d.